Theorem recexprlemex 7632

Description: 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 7633. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩

Assertion

Ref	Expression
recexprlemex	⊢ (𝐴 ∈ P → (𝐴 ·P 𝐵) = 1P)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem recexprlemex

Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩
2	1	recexprlemss1l 7630	. . 3 ⊢ (𝐴 ∈ P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
3	1	recexprlem1ssl 7628	. . 3 ⊢ (𝐴 ∈ P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))
4	2, 3	eqssd 3172	. 2 ⊢ (𝐴 ∈ P → (1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P))
5	1	recexprlemss1u 7631	. . 3 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))
6	1	recexprlem1ssu 7629	. . 3 ⊢ (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
7	5, 6	eqssd 3172	. 2 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))
8	1	recexprlempr 7627	. . . 4 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
9		mulclpr 7567	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
10	8, 9	mpdan 421	. . 3 ⊢ (𝐴 ∈ P → (𝐴 ·P 𝐵) ∈ P)
11		1pr 7549	. . 3 ⊢ 1P ∈ P
12		preqlu 7467	. . 3 ⊢ (((𝐴 ·P 𝐵) ∈ P ∧ 1P ∈ P) → ((𝐴 ·P 𝐵) = 1P ↔ ((1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P) ∧ (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))))
13	10, 11, 12	sylancl 413	. 2 ⊢ (𝐴 ∈ P → ((𝐴 ·P 𝐵) = 1P ↔ ((1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P) ∧ (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))))
14	4, 7, 13	mpbir2and 944	1 ⊢ (𝐴 ∈ P → (𝐴 ·P 𝐵) = 1P)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  {cab 2163  ⟨cop 3595   class class class wbr 4002  ‘cfv 5214  (class class class)co 5871  1^st c1st 6135  2^nd c2nd 6136  *_Qcrq 7279   <_Q cltq 7280  Pcnp 7286  1_Pc1p 7287   ·_P cmp 7289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-eprel 4288  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-ov 5874  df-oprab 5875  df-mpo 5876  df-1st 6137  df-2nd 6138  df-recs 6302  df-irdg 6367  df-1o 6413  df-2o 6414  df-oadd 6417  df-omul 6418  df-er 6531  df-ec 6533  df-qs 6537  df-ni 7299  df-pli 7300  df-mi 7301  df-lti 7302  df-plpq 7339  df-mpq 7340  df-enq 7342  df-nqqs 7343  df-plqqs 7344  df-mqqs 7345  df-1nqqs 7346  df-rq 7347  df-ltnqqs 7348  df-enq0 7419  df-nq0 7420  df-0nq0 7421  df-plq0 7422  df-mq0 7423  df-inp 7461  df-i1p 7462  df-imp 7464

This theorem is referenced by:  recexpr  7633